Temporal behaviour of global perturbations in compressible axisymmetric flows with free boundaries

Abstract

The dynamics of small global perturbations in the form of linear combination of a finite number of non-axisymmetric eigenmodes is studied in two-dimensional approximation. The background flow is assumed to be an axisymmetric perfect fluid with the adiabatic index $\gamma=5/3$ rotating with power law angular velocity distribution $\Omega \propto r^{-q}$, $1.5<q<2.0$, confined by free boundaries in the radial direction. The substantial transient growth of acoustic energy of optimized perturbations is discovered. An optimal energy growth $G$ is calculated numerically for a variety of parameters. Its value depends essentially on the perturbation azimuthal wavenumber $m$ and increases for higher values of $m$. The closer the rotation profile to the Keplerian law, the larger growth factors can be obtained but over a longer time. The highest acoustic energy increase found numerically is of order $\sim 10^2$ over $\sim 6$ typical Keplerian periods. Slow neutral eigenmodes with corotation radius beyond the outer boundary mostly contribute to the transient growth. The revealed linear temporal behaviour of perturbations may play an important role in angular momentum transfer in toroidal flows near compact relativistic objects.